Amass weighing 16 pounds stretches a spring (8/3) feet. the mass is initially released from rest from a point 2 feet below the equilibrium position, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 1/2 the instantaneous velocity. find the equation of motion if the mass is driven by an external force equal to f(t)=10cos(3t).

Answer with Step-by-step explanation:

Let a mass weighing 16 pounds stretches a spring feet.

Mass=

Mass=

Mass,m= Slug

By hook's law

A damping force is numerically equal to 1/2 the instantaneous velocity

Equation of motion :

Using this equation

Auxillary equation

Complementary function

To find the particular solution using undetermined coefficient method

This solution satisfied the equation therefore, substitute the values in the differential equation

Comparing on both sides

Adding both equation then, we get

Substitute the value of B in any equation

Particular solution,

Now, the general solution

From initial condition

x(0)=2 ft

x'(0)=0

Substitute the values t=0 and x(0)=2

Substitute x'(0)=0

Substitute the values then we get

the answer will be

this is answer of 144 cubic meters